Which equation has no solution?
$4(x+3)+2 x=6(x+2)$
$5+2(3+2 x)=x+3(x+1)$
$5(x+3)+x=4(x+3)+3$
$4+6(2+x)=2(3 x+8)$
Answer (1)
Simplify each equation.
Equation 1 simplifies to 12 = 12 , which is always true (infinitely many solutions).
Equation 2 simplifies to 11 = 3 , which is a contradiction (no solution).
Equation 3 simplifies to x = 0 (one solution).
Equation 4 simplifies to 16 = 16 , which is always true (infinitely many solutions).
Therefore, the equation with no solution is Equation 2.
The equation with no solution is 5 + 2 ( 3 + 2 x ) = x + 3 ( x + 1 ) .
Explanation
Understanding the Problem We are given four linear equations and we need to find the equation that has no solution. An equation has no solution if, after simplification, we arrive at a contradiction (e.g., 0 = 1 ).
Plan of Action Let's simplify each equation to determine if it has a solution or if it leads to a contradiction.
Simplifying Equation 1 Equation 1: 4 ( x + 3 ) + 2 x = 6 ( x + 2 )
Expanding both sides, we get 4 x + 12 + 2 x = 6 x + 12 . Combining like terms, we have 6 x + 12 = 6 x + 12 . Subtracting 6 x from both sides gives 12 = 12 , which is always true. This means the equation has infinitely many solutions.
Simplifying Equation 2 Equation 2: 5 + 2 ( 3 + 2 x ) = x + 3 ( x + 1 )
Expanding both sides, we get 5 + 6 + 4 x = x + 3 x + 3 . Combining like terms, we have 11 + 4 x = 4 x + 3 . Subtracting 4 x from both sides gives 11 = 3 , which is a contradiction. This means the equation has no solution.
Simplifying Equation 3 Equation 3: 5 ( x + 3 ) + x = 4 ( x + 3 ) + 3
Expanding both sides, we get 5 x + 15 + x = 4 x + 12 + 3 . Combining like terms, we have 6 x + 15 = 4 x + 15 . Subtracting 4 x from both sides gives 2 x + 15 = 15 . Subtracting 15 from both sides gives 2 x = 0 , so x = 0 . This equation has one solution.
Simplifying Equation 4 Equation 4: 4 + 6 ( 2 + x ) = 2 ( 3 x + 8 )
Expanding both sides, we get 4 + 12 + 6 x = 6 x + 16 . Combining like terms, we have 16 + 6 x = 6 x + 16 . Subtracting 6 x from both sides gives 16 = 16 , which is always true. This means the equation has infinitely many solutions.
Conclusion Since Equation 2 simplifies to a contradiction ( 11 = 3 ), it has no solution.
Examples
Understanding equations with no solution is crucial in various fields. For instance, in engineering, if a system of equations describing a circuit has no solution, it indicates a design flaw or an impossible configuration. Similarly, in economics, if a model predicting market equilibrium yields no solution, it suggests that the model's assumptions are inconsistent or that the market is not in equilibrium. Recognizing such situations allows professionals to identify and correct errors or adjust their models accordingly.
